Extractors Using Hardness Amplification

Abstract

Zimand [24] presented simple constructions of locally computable strong extractors whose analysis relies on the direct product theorem for one-way functions and on the Blum-Micali-Yao generator. For N-bit sources of entropy γN, his extractor has seed O(log² N) and extracts N ^γ/3 random bits.

We show that his construction can be analyzed based solely on the direct product theorem for general functions. Using the direct product theorem of Impagliazzo et al. [6], we show that Zimand’s construction can extract $\tilde \Omega_\gamma (N^{1/3})$ random bits. (As in Zimand’s construction, the seed length is O(log² N) bits.)

We also show that a simplified construction can be analyzed based solely on the XOR lemma. Using Levin’s proof of the XOR lemma [8], we provide an alternative simpler construction of a locally computable extractor with seed length O(log² N) and output length $\tilde \Omega_\gamma (N^{1/3})$.

Finally, we show that the derandomized direct product theorem of Impagliazzo and Wigderson [7] can be used to derive a locally computable extractor construction with O(logN) seed length and $\tilde \Omega (N^{1/5})$ output length. Zimand describes a construction with O(logN) seed length and $O(2^{\sqrt{\log N}})$ output length.